# Multiple Artificial Neural Networks with Interaction Noise for Estimation of Spatial Categorical Variables

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method

## 3. Case Study

#### 3.1. Synthetic Case Study

#### 3.2. Real-World Case Study

^{2}area (Figure 5). The class proportions of these five categories are (0.2046, 0.3282, 0.2432, 0.0116, 0.2124), respectively. We focus on MANN with sigmoid activation functions and consider the 10 nearest samples as the neighbors of the target point. For the proposed method, we choose the one that has the maximum correlation with output probabilities as the input and we have trained nine compositional ANNs.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Correlation plot between various transition probabilities. Q1, Q2, Q3 and Q4 denote $P\left({D}_{1}|A\right)$, $P\left({D}_{2}|A\right)$, $P\left({D}_{3}|A\right)$ and $P\left({D}_{4}|A\right)$, respectively.

**Figure 7.**Training errors in the iteration process of MANN with interaction noise; different colors and line styles are used to distinguish the nine compositional ANNs.

**Figure 8.**(

**a**) Original image obtained from

**R**software data set (jura.grid); (

**b**) Classification maps generated by MANN with interaction noise. Simulations are conditioned on 359 samples in Jura lithology data set, note the small Quaternary patches (visualized in black) neglected in (a) are successfully recovered in the result of the proposed method (b).

**Table 1.**Prediction accuracy comparison of the raster data set. $\mu $ and $\sigma $ denote the mean and standard deviation, respectively.

Parameters | Class 1 | Class 2 | Class 3 | Total |
---|---|---|---|---|

$\mu =0$;$\sigma =0.1$ | 58.02% (579/998) | 50.49% (830/1644) | 60.97% (517/848) | 55.19% (1926/3490) |

$\mu =-1$;$\sigma =0.1$ | 58.62% (585/998) | 50.67% (833/1644) | 60.85% (516/848) | 55.42% (1934/3490) |

$\mu =-1$;$\sigma =0.5$ | 58.82% (587/998) | 51.03% (839/1644) | 60.02% (509/848) | 55.44% (1935/3490) |

$\mu =-2$;$\sigma =0.5$ | 57.72% (576/998) | 52.86% (869/1644) | 59.55% (505/848) | 55.87% (1950/3490) |

$\mu =-3$;$\sigma =0.9$ | 54.91% (548/998) | 56.20% (924/1644) | 57.90% (491/848) | 56.25% (1963/3490) |

$\mu =-3$;$\sigma =1.2$ | 55.71% (556/998) | 55.90% (919/1644) | 58.37% (495/848) | 56.45% (1970/3490) |

$\mu =-4$;$\sigma =1.2$ | 52.30% (522/998) | 57.85% (951/1644) | 56.60% (480/848) | 55.96% (1953/3490) |

$\mu =-4$;$\sigma =1.5$ | 53.51% (534/998) | 57.24% (941/1644) | 57.19% (485/848) | 56.16% (1960/3490) |

$\mu =-5$;$\sigma =1.2$ | 53.51% (518/998) | 54.01% (888/1644) | 57.08% (484/848) | 54.15% (1890/3490) |

$\mu =-5$;$\sigma =1.5$ | 50.00% (499/998) | 55.41% (911/1644) | 56.84% (482/848) | 54.21% (1892/3490) |

MCRF | 57.21% (571/998) | 51.64% (849/1644) | 60.85% (516/848) | 55.47% (1936/3490) |

**Table 2.**Training weights of MANN with interaction noise. Input node, hidden nodes and output node are represented by i, h1 to h3 and o, respectively; b1 and b2 are the bias nodes in the hidden layers.

Nodes | Weights | ||||||||
---|---|---|---|---|---|---|---|---|---|

ANN1 | ANN2 | ANN3 | ANN4 | ANN5 | ANN6 | ANN7 | ANN8 | ANN9 | |

b1->h1 | 0.11 | −3.74 | −2.15 | −0.15 | −0.11 | 5.61 | −0.05 | 0.23 | 0.03 |

i->h1 | 0.66 | 5.64 | 2.54 | −6.28 | 2.04 | −7.38 | 1.97 | −2.48 | −0.54 |

b1->h2 | 0.03 | 0.65 | −0.53 | 0.29 | 4.02 | 0.56 | 6.08 | 0.24 | 0.53 |

i->h2 | 0.14 | 0.52 | 1.08 | −1.02 | −4.93 | 2.65 | −7.00 | −2.48 | −0.42 |

b1->h3 | −5.43 | 0.66 | 1.04 | −3.16 | 0.60 | 0.09 | 0.28 | 0.24 | 2.73 |

i->h3 | 6.00 | 10.09 | −1.58 | 3.85 | 12.06 | 0.13 | 0.80 | −2.48 | −5.02 |

b2->o | −0.49 | 2.31 | −0.70 | −0.21 | 4.87 | 0.91 | 1.42 | 2.33 | 0.55 |

h1->o | −0.75 | 4.81 | 3.05 | −4.61 | 4.83 | −4.12 | 2.18 | −2.87 | 0.62 |

h2->o | −0.24 | 1.71 | 1.18 | −1.24 | −3.58 | 2.25 | −4.72 | −2.87 | 0.60 |

h3->o | 6.76 | −5.56 | −2.32 | 4.12 | −5.68 | 0.47 | 1.30 | −2.87 | -2.91 |

Method | Argovian | Kimmeridgian | Sequanian | Portlandian | Quaternary | Total |
---|---|---|---|---|---|---|

MANN with interaction noise | 75.78% (898/1185) | 78.19% (1592/2036) | 64.07% (1043/1628) | 0.00% (0/316) | 43.69% (346/792) | 65.12% (3879/5957) |

MCRF | 72.07% (854/1185) | 74.17% (1510/2036) | 61.61% (1003/1628) | 0.00% (0/316) | 47.60% (377/792) | 62.85% (3744/5957) |

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**MDPI and ACS Style**

Huang, X.; Wang, Z. Multiple Artificial Neural Networks with Interaction Noise for Estimation of Spatial Categorical Variables. *Algorithms* **2016**, *9*, 56.
https://doi.org/10.3390/a9030056

**AMA Style**

Huang X, Wang Z. Multiple Artificial Neural Networks with Interaction Noise for Estimation of Spatial Categorical Variables. *Algorithms*. 2016; 9(3):56.
https://doi.org/10.3390/a9030056

**Chicago/Turabian Style**

Huang, Xiang, and Zhizhong Wang. 2016. "Multiple Artificial Neural Networks with Interaction Noise for Estimation of Spatial Categorical Variables" *Algorithms* 9, no. 3: 56.
https://doi.org/10.3390/a9030056